L(s) = 1 | − 1.23·3-s − 3.23·5-s + 7-s − 1.47·9-s + 6.47·11-s − 0.763·13-s + 4.00·15-s + 4.47·17-s + 1.23·19-s − 1.23·21-s + 4·23-s + 5.47·25-s + 5.52·27-s + 4.47·29-s − 2.47·31-s − 8.00·33-s − 3.23·35-s + 4.47·37-s + 0.944·39-s − 8.47·41-s − 6.47·43-s + 4.76·45-s + 10.4·47-s + 49-s − 5.52·51-s + 10·53-s − 20.9·55-s + ⋯ |
L(s) = 1 | − 0.713·3-s − 1.44·5-s + 0.377·7-s − 0.490·9-s + 1.95·11-s − 0.211·13-s + 1.03·15-s + 1.08·17-s + 0.283·19-s − 0.269·21-s + 0.834·23-s + 1.09·25-s + 1.06·27-s + 0.830·29-s − 0.444·31-s − 1.39·33-s − 0.546·35-s + 0.735·37-s + 0.151·39-s − 1.32·41-s − 0.986·43-s + 0.710·45-s + 1.52·47-s + 0.142·49-s − 0.774·51-s + 1.37·53-s − 2.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9209027650\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9209027650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.40447279000508985598442576999, −10.46069723831881092012825615668, −9.154303778656024394519853196568, −8.366086263278235606408736912743, −7.35129617628906922001146181175, −6.51146798637493478259712053812, −5.31296608222213756198279977427, −4.24185804371242371066282270128, −3.29794345337559195465054323068, −0.976012793566189189926165323570,
0.976012793566189189926165323570, 3.29794345337559195465054323068, 4.24185804371242371066282270128, 5.31296608222213756198279977427, 6.51146798637493478259712053812, 7.35129617628906922001146181175, 8.366086263278235606408736912743, 9.154303778656024394519853196568, 10.46069723831881092012825615668, 11.40447279000508985598442576999